HEXAGONbeta

Records, scale flows, and Gaussian limits from lattice Poisson fields

Primarily AI-generated textHuman understanding: no partsmath.PR — Probabilitymath.NT — Number Theory

Contributed by Brice Pouly ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

How does a rare-event field determine the history of its extremes? Starting from the complete signed run-field comparison in Long runs and rare patterns, we develop records, censoring, occupation and Gaussian limits for the resulting lattice Poisson target. The arithmetic input retains every site, full exact length and actual sign, at growing intensity under a one-factor information-intensity budget. Strict integer records, their signed imbalance and logarithmic leadership time have a joint phase-uniform Brownian limit with explicit covariance, independent of the terminal critical field and its extremal paths. Right censoring gives scalar M1M_1 convergence for the visible maximum and J1J_1 convergence for the first-holder sign. Physical occupation instead has a nondegenerate phase-indexed limit. Canonical interpolation connects threshold thinning and a stationary scale flow without identifying their record rules or their Gaussian constants. The signed spatial theory includes weighted exact levels, fixed-window Ornstein-Uhlenbeck paths, two correlated threshold-compression noises, and rectangle fields with explicit grid centering. Direct partition and quadratic-reconstruction proofs provide alternative routes. For each specified observation, the target calculations identify its limit and the dependencies within the joint constructions.

Provenance statement

The following declaration describes the research and the published source edition: The author directed this research with substantial assistance from Claude Fable 5 and Claude Opus 4.8 (Anthropic), and GPT-5.6 Sol and GPT-6 Astra (OpenAI). Assistance covered candidate arguments, mathematical drafting and reorganization, reference checks, and executable diagnostics. GPT-6 Astra also assisted with the signed spatial hierarchy, the arithmetic-interface developments and their integration, and the editorial and bibliographic preparation. The author determines the claims and scope and accepts responsibility for the arguments, references, and remaining errors. AI-assisted checks are not independent human peer review. The claims rest on the printed proofs and stated inputs, not on model outputs or finite tests. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.22876520. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

References

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026